Quantifying Quantum Coherence in Polariton Condensates

We theoretically and experimentally investigate quantum features of an interacting light-matter system from a multidisciplinary perspective, combining approaches from semiconductor physics, quantum optics, and quantum-information science. To this end, we quantify the amount of quantum coherence that results from the quantum superposition of Fock states, constituting a measure of the resourcefulness of the produced state for modern quantum protocols. This notion of quantum coherence from quantum-information theory is distinct from other quantifiers of nonclassicality that have previously been applied to condensed-matter systems. As an archetypal example of a hybrid light-matter interface, we study a polariton condensate and implement a numerical model to predict its properties. Our simulation is confirmed by our proof-of-concept experiment in which we measure and analyze the phase-space distributions of the emitted light. Specifically, we drive a polariton microcavity across the condensation threshold and observe the transition from an incoherent thermal state to a coherent state in the emission, thus confirming the buildup of quantum coherence in the condensate itself.

Popular Summary

The next information technology revolution is the quantum computer, promising a performance boost that exceeds any classical devices by exploiting the peculiar laws of quantum physics. However, connecting different components of quantum devices, such as matter systems that process quantum information and quantum light that transports this information, presents a major challenge when attempting to realize fully operational and useful quantum technologies. In this work, a new approach is put forward that interconnects such seemingly incompatible quantum systems. In particular, genuine quantum superposition effects in Bose-Einstein condensates are probed by characterizing the coupled quantum light fields.

By combining quantum resource theories from quantum-information science, state-of-the-art numerical simulations of semiconductor microcavities, and unique quantum-optical experiments, it is rendered possible to assess the amount of quantum coherence from the superposition of polaritons, the quasiparticles of a hybrid light-matter system. In contrast to previous methods, such quantum features are studied by measuring photon superpositions in the emitted light, thereby confirming quantum coherence—a resource for quantum-information processing—within the generated polariton condensate. This multidisciplinary research connects different fields and arches across different physical platforms, which is essential for the future application and success of quantum devices. At the same time, fundamental superposition effects can be studied in this manner, paving the way for discovering new quantum phenomena in hybrid quantum systems.

Figure 1

Conceptual goal of our interdisciplinary research. A matter system represents a quantum device, e.g., a quantum processor, in which quantum superpositions are produced. A light field is used for distributing the generated quantum coherence, e.g., for quantum communications. The light-matter interaction serves as an interface. By analyzing the quantum features of the light field, quantum coherence of the semiconductor system can be characterized to assess the system’s resourcefulness for quantum-information protocols.

Figure 2

Quantum coherence $\mathcal{C}(\hat{\rho })$ as a function of the coherent and thermal photon numbers, $0\le |{\alpha }_0{|}^2\le 10$ and $0\le \overline{n}\le 10$, respectively. With increasing displacement ${\alpha }_0$, the amount of quantum coherence increases. And a reduction of coherence is observed when the thermal background $\overline{n}$ increases.

Figure 3

(a) Sketch of a semiconductor microcavity consisting of a quantum well (QW) that is embedded inside a cavity, realized by distributed Bragg reflectors (DBR). (b) Upper and lower polariton branch (UPB and LPB, respectively) alongside the bare cavity and exciton dispersions. A nonresonant pump injects free carriers (top, red), which form excitons and relax down to the LPB, building up a reservoir (left and right, yellow). From the reservoir, stimulated scattering results in the formation of the polariton condensate (bottom, blue) near the zero-momentum ground state.

Figure 4

Simulated first-order, equal-time spatial coherence as a function of the distance from the center of the excitation spot for increasing pump intensities, evidencing the creation of phase coherence across the excitation spot.

Figure 5

Results from numerical simulations. (a) Single-time snapshot of the $k$-space density expectation values above the condensation threshold, $P=2P_{\mathrm{thr}}$. The red square indicates the selected signal area for the mode occupation average. (b) Mean and (c) variance of the averaged polariton excitation number as a function of the pump intensity. (c) Equal-time, second-order correlation function $g^{(2)}(\tau =0)$ as a function of the pump intensity, showing the transition from a thermal state to a coherent state across the condensation threshold. (d) Resulting amount of quantum coherence $\mathcal{C}$.

Figure 6

Experimental setup. The signal that is emitted by the polariton microcavity is split into two channels. Each channel is interfered with the local oscillator (LO), before being detected by a balanced detector (BD). The LO path of channel 1 contains a piezo mirror (Pz) for sweeping the relative phase. The two BDs yield measurements of the quadratures $X_1$ and $X_2$ that are postselected for retrieving orthogonal quadratures $q$ and $p$, and that are then binned to create a histogram $Q(q,p)$, which corresponds to the Husimi function. (PBS, polarizing beam splitter; GT, Glan-Thompson prism; $\lambda /2$, half-wave plate.)

Figure 7

Reconstructed phase-averaged Husimi function (a) for $P=P_{\mathrm{thr}}$ and (b) for $P=7.5P_{\mathrm{thr}}$. The bottom plots depict cuts along the $q$ axis ($p=0$). Black lines represent data while red curves show the fitted function of a displaced thermal state. A one standard-deviation error margin, given by the standard error of the counting statistics, is shown as shaded areas but is mostly not visible.

Figure 8

Time-resolved photon number and $g^{(2)}(\tau =0,t)$ for $P=6.8P_{\mathrm{thr}}$. The initial time $t=0$ is arbitrarily chosen as the beginning of the measurement. A dashed line in the bottom panel marks the frontier between the state with high intensity and the one with low intensity. Right panels show the phase-averaged Husimi functions above (top plot) and below (bottom plot) this limit.

Figure 9

Results derived from fitting phase-averaged Husimi functions as a function of the pump intensity for vertical polarization. (a) Coherent (red circles) and thermal (black circles) photon numbers. Black line shows the total photon number $n_{\mathrm{total}}$. Thick, black vertical lines indicate thresholds between different regimes of emission. (b) Ratio between the coherent and thermal photon-number contribution. (c) Equal-time, second-order correlation function $g^{(2)}(\tau =0)$. (d) Amount of quantum coherence. Open and closed symbols correspond to the low and high state, respectively, when the emission is switching on and off. When both symbols overlap, the emission is stable. Error bars are obtained from a Monte Carlo error propagation (see Appendix pp5 for details); because of the asymmetry of the logarithmic scale, only the upper part of the error margin is depicted.

Figure 10

Dispersion curves measured at different excitation powers. In each panel, the emission is normalized to one and plotted on a linear color scale. The red line shows the lower polariton dispersion obtained from fitting the dispersion at a power below threshold. In plot (a) at $P_{\mathrm{thr}}$ and in (b) at $1.3P_{\mathrm{thr}}$, the emission around $k=0$ is blueshifted because of interaction with the reservoir. In (b), emission at $k=\pm 0.8\mu {\mathrm{m}}^{-1}$ is visible. In (c), $7.5P_{\mathrm{thr}}$, this side emission is mostly suppressed when compared with the dominant emission at $k=0$, which is redshifted because of heating.